Question: The triangle below is equilateral. Find the length of side $x$ in simplest radical form with a rational denominator. $\sqrt{10}$ $x$ Answer Graph: An equilateral triangle is shown near the center of the page, with the left side labeled $\sqrt{10}$ and an interior segment from the bottom vertex to the top side labeled $x$; the interior segment is perpendicular to the top side, so position_hint \in \{center\}.

1. **State the problem:** We have an equilateral triangle with one side length $\sqrt{10}$ and a perpendicular segment $x$ from the bottom vertex to the opposite side. We need to find $x$ in simplest radical form with a rational denominator. 2. **Recall properties of equilateral triangles:** All sides are equal, so each side is $\sqrt{10}$. The perpendicular segment from a vertex to the opposite side is the altitude. 3. **Formula for altitude in an equilateral triangle:** The altitude $h$ of an equilateral triangle with side length $s$ is given by $$h = \frac{\sqrt{3}}{2} s$$ 4. **Apply the formula:** Substitute $s = \sqrt{10}$: $$h = \frac{\sqrt{3}}{2} \times \sqrt{10} = \frac{\sqrt{3} \times \sqrt{10}}{2} = \frac{\sqrt{30}}{2}$$ 5. **Simplify the expression:** The denominator is already rational, so the simplest radical form with a rational denominator is $$x = \frac{\sqrt{30}}{2}$$ **Final answer:** $$\boxed{\frac{\sqrt{30}}{2}}$$